A current I is flowing along the y-axis and a spherical surface with radius 1 m

Question

A current I is flowing along the y-axis and a
spherical surface with radius 1 m has its center at
origin, as in the figure left. A closed contour C is
chosen as in the figure, which is a boundary
between two semi-sphere surfaces S1 and S2. Based
Ion the uniqueness of magnetic field circulation
C
H¢ dl
I
calculated from both surfaces S1 and S2,
find the total displacement current emanating from
the spherical surface using the Ampere’s law.
(HINT: Be careful with the normal directions of
the surfaces and right-hand relationships)

Explanation / Answer

Well, maybe I can at least give you an idea or two. First, current i must be time-varying, or nothing happens electric-field-wise. Second: using Ampere, write an expression for B around the wire, including the space defined by the two hemispheres. BTW the hemispheres are just geometrically descriptive surfaces. They have no material meaning. Least that's what I assume. Third: now you have B(x,y,z). What is the Maxwell relation that relates E to B? Fourth: what is the relation between E and D? Assume non-conducting medium. Fifth: What is the meaning of "displacement current emanating from the spherical surface", given D and the surfaces? Think of an analogy with how you get from conduction current density J to conduction current in the Maxwell relation relating B to J and D. Sixth: how can you apply Stokes' theorem in conjunction with item 3 to calculate item five?

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